Babylonian numerals are a place value system with base 60, the sexagesimal system. They need only two cuneiform signs: a vertical wedge for 1 and a corner wedge for 10. From these you build every digit from 1 to 59. As in our decimal system, the value of a digit depends on its place, except that each place is worth 60 times the place to its right.
Below you learn the signs, the chart from 1 to 59, how to read and write the places, how to convert in both directions and how to calculate with sexagesimal fractions. You also see why the Babylonians wrote no comma and for a long time no zero, and why we count 60 minutes in an hour and 360 degrees in a circle. The history is told on the page history of mathematics, and a German version is at Babylonische Zahlen (in German).
The two cuneiform signs: wedge and corner wedge
The scribes pressed a reed stylus into wet clay. This made wedge-shaped marks, which is where the name cuneiform (from Latin cuneus, wedge) comes from. A vertical wedge means 1 and a corner wedge (it looks like a sideways arrow head) means 10. For a digit you put the signs together: 23 is 2 corner wedges and 3 wedges.
Babylonian numerals 1 to 59: the chart
The chart shows every Babylonian digit from 1 to 59. Each row has ten numbers, so the numbers in a column share the same wedges and the corner wedges increase by one at every new row. The signs are drawn in a simplified way.
Place value in base 60
Larger numbers are written with several places next to each other. The highest place is on the left and the ones on the right, as with us. Each place is 60 times the place to its right, so the place values are the powers of 60.
| Place (from the right) | Value | As a power |
|---|---|---|
| 1 | 1 | 600 |
| 2 | 60 | 601 |
| 3 | 3,600 | 602 |
| 4 | 216,000 | 603 |
| 5 | 12,960,000 | 604 |
Today the places are written with commas: 1,23 means 1 sixty and 23 ones. On the tablets there was no comma, only the signs. The gap between places showed where a new one began.
How to convert Babylonian numerals
From Babylonian numerals to our numbers
- Read each place as a digit: 10 times the corner wedges plus the wedges.
- Write the place value beside it: from the right 1, 60, 3,600, and so on.
- Multiply every digit by its place value and add.
Example: the places 12,40 are 12 · 60 + 40 = 720 + 40 = 760.
From our numbers to Babylonian numerals
Divide the number by 60 again and again and note the remainders. The first remainder is the place on the far right, the next is the place to its left. Stop when the quotient is 0. It is the same method as for binary numbers and Mayan numbers, only with 60. Write every remainder as a digit made of corner wedges and wedges.
| Calculation | Remainder | Place (from the right) |
|---|---|---|
| 2,026 : 60 = 33 | 46 | 1 |
| 33 : 60 = 0 | 33 | 2 |
The remainders 46 and 33 give 33,46. Check: 33 · 60 + 46 = 2,026.
| Decimal | Base 60 digits | Calculation |
|---|---|---|
| 59 | 59 | 59 |
| 60 | 1,0 | 1 · 60 |
| 61 | 1,1 | 1 · 60 + 1 |
| 100 | 1,40 | 1 · 60 + 40 |
| 120 | 2,0 | 2 · 60 |
| 3,600 | 1,0,0 | 1 · 3,600 |
| 4,000 | 1,6,40 | 1 · 3,600 + 6 · 60 + 40 |
| 12,340 | 3,25,40 | 3 · 3,600 + 25 · 60 + 40 |
| 2,026 | 33,46 | 33 · 60 + 46 |
| 7,205 | 2,0,5 | 2 · 3,600 + 5 |
The number system converter (interface in German) shows how the Babylonians would have written any number you enter in cuneiform.
No zero and no comma: the context decides
The Babylonians wrote neither a comma nor a zero. In the Old Babylonian period an empty place was left as a small gap, which was easy to overlook. They also did not mark where the ones are. So the same sign sequence “1 30” could mean different things.
| Written | Calculation | Value |
|---|---|---|
| 1,30 | 1 · 60 + 30 | 90 |
| 1;30 | 1 + 30/60 | 1½ |
| 1,30,0 | 1 · 3,600 + 30 · 60 + 0 | 5,400 |
Which value was meant came from the problem. For a field area or a price, a trained scribe knew the right order of magnitude. Only in the 1st millennium BC did a placeholder for empty places appear. It usually stood between two digits, and nobody calculated with it.
Sexagesimal fractions: calculating without remainder
The Babylonians wrote fractions in base 60 as well. After the semicolon come sixtieths, then 3,600ths and so on: 0;30 is 30/60, that is 1/2. The semicolon is a modern device that was not on the tablets. Because 60 has so many divisors, many fractions end exactly. For thirds you see the advantage: 1/3 is 0;20, but in decimal 0.333… without end.
| Fraction | Sexagesimal | Decimal (rounded) |
|---|---|---|
| 1/2 | 0;30 | 0.5 |
| 1/3 | 0;20 | ≈ 0.3333 |
| 1/4 | 0;15 | 0.25 |
| 1/5 | 0;12 | 0.2 |
| 1/6 | 0;10 | ≈ 0.1667 |
| 1/8 | 0;7,30 | 0.125 |
| 1/9 | 0;6,40 | ≈ 0.1111 |
| 1/12 | 0;5 | ≈ 0.0833 |
| 1/15 | 0;4 | ≈ 0.0667 |
| 1/16 | 0;3,45 | 0.0625 |
| 1/20 | 0;3 | 0.05 |
| 1/25 | 0;2,24 | 0.04 |
| 1/27 | 0;2,13,20 | ≈ 0.0370 |
The Babylonians divided with tables of reciprocals: they computed a : b as a · (1/b). The numbers 7, 11 and 13 are missing from the tables, because their reciprocals do not end. A sexagesimal fraction ends only if the denominator has no prime factors other than 2, 3 and 5. The fraction 1/7 gives 0;8,34,17,8,34,17 and so on forever.
Why 60? Minutes, seconds and 360 degrees
Why the Sumerians and Babylonians chose 60 is not recorded. A strong argument is that it divides so easily. The number 60 has twelve divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. The number 10 has only four. 60 is also the smallest number that 1, 2, 3, 4, 5 and 6 all divide into. Half, a third, a quarter, a fifth and a sixth of an hour are therefore whole minutes: 30, 20, 15, 12 and 10.
The sexagesimal system lives on in time and angles. An hour has 60 minutes and a minute has 60 seconds. A degree is divided into 60 arc minutes, and an arc minute into 60 arc seconds. The words “minute” and “second” come from Latin pars minuta prima and pars minuta secunda, the first and the second small part, meaning the first and the second sixtieths.
Greek astronomers took over the sexagesimal fractions. Ptolemy calculated with them throughout the Almagest, and Hipparchus relied on Babylonian observations. That a full circle has 360 degrees is 6 · 60. It is certain that Babylonian astronomers divided the path of the Sun into twelve zodiac signs of 30 parts each. But this is not certainly the only origin of the 360 degrees: other candidates are the roughly 360 days of a year and the many divisors of 360, which has 24.
Common mistakes
Practice problems with answers
Where Babylonian numerals come from
The first number signs of the Sumerians arose around 3300 to 3200 BC and had no place value yet. The place value system in base 60 came into the scribes’ schools by the Old Babylonian period at the latest, around 2000 BC. From that time come the tablets with multiplications, reciprocals and equations, among them Plimpton 322 and YBC 7289. A different place value system is shown by the Mayan numbers in base 20, and a system without place value by Egyptian numerals. How these systems fit together is in number systems.
Frequently asked questions
How does the Babylonian number system work?
It has two signs, the wedge for 1 and the corner wedge for 10, and base 60. One place shows 1 to 59, and each place is worth 60 times the place to its right. Examples are in place value in base 60.
How do you convert Babylonian numerals?
From Babylonian: multiply each place by its value 1, 60, 3,600 and add. To Babylonian: divide by 60 again and again and write the remainders from right to left as places. See converting.
What is the sexagesimal system in simple terms?
A number system with base 60 in which each place is worth 60 times the place to its right. It comes from Mesopotamia and lives on in the clock (60 minutes, 60 seconds) and in angles.
Did the Babylonians have a zero?
Not as a number. Later there was a placeholder for empty places, but nobody calculated with it. Zero as a number came from India.
Why does an hour have 60 minutes?
The division goes back to the sexagesimal system of the Babylonians, which Greek astronomers took over. Since 60 has many divisors, half, a third, a quarter and a sixth of an hour are whole minutes.
What are the Babylonian numerals from 1 to 59?
They are made from vertical wedges (ones, up to 9) and corner wedges (tens, up to 5). The chart shows all of them.
Why was 60 chosen as the base?
The reason is not recorded. The strongest argument is the many divisors of 60. The popular idea of counting finger joints is not documented. See why 60.
Sources and further reading
- MacTutor History of Mathematics: Babylonian numerals
- MacTutor History of Mathematics: Pythagoras’s theorem in Babylonian mathematics
- Eleanor Robson: Mathematics in Ancient Iraq. A Social History, Princeton University Press, Princeton 2008.
- Otto Neugebauer: The Exact Sciences in Antiquity, 2nd edition, Brown University Press, Providence 1957.
- Jens Høyrup: Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and Its Kin, Springer, New York 2002.
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